Question

Let `alpha` and `beta` be the roots of `x^2-6x-2=0` with `alpha>beta` if `a_n=alpha^n-beta^n` for ` n>=1` then the value of `(a_10 - 2a_8)/(2a_9)`
A. 1
B. 2
C. 3
D. 4

Answer 1

Correct Answer - C
`(a_(10)-2a_(8))/(2a_(8))=((alpha^(10)-beta^(10))-2(alpha^(8)-beta^(8)))/(2(alpha^(9)-beta^(9)))`
`=(alpha^(8)(alpha^(2)-2)-beta^(8)beta^(2)-2)/(2(alpha^(9)-beta^(9)))`
`because alpha"is root of"x^(2)-6x2=0impliesalpha^(2)-2=6alpha[andbeta"is root of"x^(2)-6x-2=0impliesbeta^(2)-2=6beta]`
`=(alpha^(8)(6 alpha)-beta^(8)(6beta))/(2(alpha^(9)-beta^(9)))=(6(alpha^(9)-beta^(9)))/(2(alpha^(9)-beta^(9)))=3`